High dimensional expanders and Ramanujan complexes

Alex Lubotzky, Weizmann Institute, The Hebrew University, and Minerva Distinguished Visitor
Fine Hall Common Room

The seminal work of Howard Garland in the 60's, proving the conjecture of Serre on vanishing cohomology of lattices in $p$-adic simple Lie groups reveals that high dimensional expanders (HDX) have a ``local to global" phenomenon which does not exist in graphs. Furthermore, the work of L. Lafforgue, generalizing Drinfeld's work from $GL(2)$ to $GL(n)$, called for the construction of Ramanujan complexes. These high dimensional simplicial complexes led to the development of a rich theory with applications in CS and math.